Free Energy, Vapor Pressure and
the Equilibrium Between a Vapor
and Condensed Phase
References:
Thermodynamics, G. N. Lewis and M. Randall,
rev. K. Pitzer and L. Brewer, McGraw Hill, New
York (1961).
Chemical Thermodynamics, I. Klotz, Benjamin,
New York (1964).
• The Gibbs Free Energy is generally
agreed to be the “weapon of choice” for
describing (a) chemical reactions and (b)
equilibria between phases. It is defined
as:
•
G = H – TS = E + PV – TS
(1)
• Where H = Enthalpy
•
E = Total internal energy
•
T = [Absolute] Temperature
•
S = Entropy
• Obviously dG = dE + PdV +VdP – TdS – SdT
Aside
• Remember that thermodynamic variables
come in pairs
• One is “intrinsic” (does not depend on
system size)
• One is “extrinsic” (depends on system
size)
• Examples: P and V, T and S…
• Also G and N, the number of moles of stuff
in the system.
• Hence G is the appropriate variable when
material is moving between phases
From the
First Law of Thermodynamics
• dE = TdS – PdV since dS = δQ/T and the
mechanical work done on a system when
it expands is –PdV.
• Substituting into
• dG = dE + PdV +VdP – TdS – SdT
• Leaves:
dG = -SdT + VdP
Apply equation to a closed system
• Closed system contains pure substance
– vapor
– condensed phase
• Phases co-exist in equilibrium.
• Objective will be to see how the pressure
of the vapor depends on the temperature
of the system.
Write the Free Energy Equation twice
• Once for each phase
• dGc = -ScdT + VcdP
condensed phase
• dGv = -SvdT + VvdP
vapor phase
c refers to the
v refers to the
Objective of the Exercise
• Change system temperature
• See how the pressure of the vapor
changes
• Note:
– Both the vapor and condensed phase are in
thermal equilibrium
– Both are at the same temperature
– System remains in thermal equilibrium if the
molar free energy of the system, also known
as the chemical potential, remains constant.
Definition of chemical equilibrium
between two phases
• Free energy is the same in both phases
• Gc = Gv
• Changes in free energy when some
independent variable is changed must be
the same if they are to remain in
equilibrium
dGc = dGv
• -ScdT + VcdP = -SvdT + VvdP
• (Sv - Sc )dT = (Vv- Vc)dP
• (Sv - Sc ) is the entropy change that takes
place when material moves from the
condensed phase to the vapor
• ΔS = ΔQ/T where ΔQ is the amount of
heat required per mole of material moved
between the phases
• ΔQ is just the heat of vaporization!
• dP/dT = (Sv – Sc)/(Vv – Vc) = ΔHv/(TΔV)
• This is the Clapeyron equation (E.
Clapeyron, J. ẻcole polytech. (Paris)
14(23), 153 (1834). It relates the change
in pressure of a vapor to the temperature
in a closed, mono-component system to
the heat of vaporization, system
temperature and molar volume change of
the material on vaporization.
Creating of an Ideal Gas
• For lack of a better model, we treat most
vapors as ideal gases, whose molar
volume is given by:
•
V/n = RT/P
• Alternatively, equation of state is needed
• Molar volume of gas is typically factor of
500 larger than condensed phase
• Hence Vc is negligible in comparison
Substituting and Integrating
• dP = (ΔHv/Vv)dT/T = (PΔHv/RT)dT/T
• dP/P = ΔHv/R)dT/T2
• ln(P(T)/ P0) = -(ΔHv/R)(1/T – 1/T0)
• P(T) = P0 exp(-ΔHv/R(1/T – 1/T0))
• The vapor pressure in equilibrium with a
condensed phase increases exponentially
(sort of: exp(-1/T) isn’t exactly an
exponential!) with temperature from zero
up to the critical temperature.
• Borne out in the vapor pressure charts and
generating functions for them.
• Deviations from linearity on the log-log plot
– Temperature dependence of the heat of
vaporization
– exp (-1/T) isn’t really linear in the exponent.
Heat of Vaporization from CRC Data
Log10p(Torr) = -0.2185*A/T + B
Vapor Pressure of Water
Vapor Pressure (Torr)
10000
"Normal boiling point"
1000
100
10
1
0.1
-20
0
20
40
60
Temperature (C)
80
100
120
Generating Function s Imperfect
• Normal boiling point of water on the plot is
about 108.2 C.
• I cheated; I looked at the data!
• What does “Normal boiling point” mean??
• What does “boiling” mean???
Kubaschewski, Evans and Alcock, Metallurgical
Thermochemistry, Pergammon, Oxford (1967)
• Log10p(Torr) = A/T + BlogT + CT + D
Vapor Pressure of Water
Vapor Pressure (Torr)
10000
"Normal boiling point"
1000
100
10
Kubaschewski et al.
1
0.1
-20
0
20
40
60
Temperature (C)
80
100
120
From Kubaschewski’s Data
•
•
•
•
Vapor pressure at 100 C is 758 Torr
Much better approximation!
Caveat emptor!
“Life is like a sewer: the quality of the fit
depends on the number of terms in the
generating function!”
RCA Charts
• Great resource tool: the “RCA Charts”,
which live in 327 JFB. Treat them with
great respect: they are much older than
you are and irreplaceable.
• “Often imitated
• Never duplicated”
Other Sources of Data
• Vapor Pressures of Pure Substances,
Daniel R. Stull, Industrial and Engineering
Chemistry, April 1947, pp. 517 – 550.
(Tables give data on temperature required
to generate a specific vapor pressure.)
• Vapour Pressure of the Elements, An. N.
Nesmeyanov, Academic Press, NY, NY
(1963).
Gibbs Phase Rule
•
•
•
•
•
•
•
F=C–P+2
F = number of degrees of freedom
C = number of components
P = number of phases
What is a phase?
What is a component?
What is a degree of freedom?
Phase Diagram of Water
Melting
curve
Boiling curve
“Normal boiling point”
“Melting
point”
Sublimation curve
Homework
1. Determine the vapor pressure at 77 K
– Water
– Carbon monoxide
2. What is the boiling point of water in a
vacuum system at 10-6 Torr?
3. What is the vapor pressure of carbon
dioxide at room temperature?
4. What is the condensed phase of carbon
dioxide in equilibrium with the vapor at
room temperature? How do you know?